Download Cardiac Muscle and Organ Mechanics

Cardiac Muscle and
Organ Mechanics
Roy Kerckhoffs
Dept of Bioengineering,
University of California, San Diego
Tutorial on heart and lungs
Ohio State University,
Columbus, OH, 20 sep 2006
Outline
system
system
organ
organ
tissue
tissue
cell
cell
kn
Ca2+
b
Ca2+
R*offk
k*of
f
kon
R*on
g f
A*1
R0off
Ca2+
R0on
Ca2+
g
Ca2+
A01
Ca2+
Overview
• Anatomy and physiology of the heart
– System and organ level
– Resting cardiac tissue
– Active force generation: the sarcomere
• Multi-scale modeling
– Anatomic models
– Models of cardiac mechanics: from cell to system
– Models of cardiac electromechanics
Cardiac Anatomy
Mitral
SVC Aorta
Pulmonic
Pulmonary
artery
Tricuspid
valves
LA
RA
LV
RV
Septum
Epicardium
Endocardium
Apex
Base
Vperi
Pericardium
• A sac wherein the heart sits
• Limits sudden increases in volume
• Increases atrio-ventricular and ventricularventricular interaction
Vaw
Vra
Vla
Vrv
Vlv
Vvw
*Freeman & Little, AJP 1986;251:H421-H427
Physiology
Conduction
of System
the heart
right atrium
sinusnode
AV-node
right
bundle
branch
left atrium
left bundle branch
left ventricle
Purkinje fibers
right ventricle
Coronary
System
2
The Cardiac
Cycle
1
3
4b
4a
Systole:
1. Isovolumic
contraction
2. Ejection
4a
4b
Diastole:
3. Relaxation
Early
Isovolumic
4. Filling
a) Early, rapid
b) Late, diastasis
1
3
2
3
4a
16
AVC
Pressure (kPa)
14
12
Aorta
AVO
10
8
6
4
Left ventricle
2
MVO
MVC
0
150
Volume (ml)
Pressure
and Volume
2
1
4b
120
90
60
30
0
100
200
300
400
Time (msec)
500
600
700
The Pressure-Volume Diagram
Endsystole
(ES)
SV=EDV-ESV
Ejection
16
AVC
Ejection Fraction
EF=SV/EDV
AVO
Stroke
volume
(SV)
8
Isovolumic
contraction
12
Isovolumic
relaxation
Pressure (kPa)
20
End-diastole
(ED)
4
MVO
Filling
MVC
0
0
50
100
Volume (ml)
150
200
The Pressure-Volume Diagram
20
ESV
EW 
16
 P(t )d V
EDV
AVC
AVO
8
Stroke
(external)
work
Isovolumic
contraction
12
Isovolumic
relaxation
Pressure (kPa)
Ejection
4
MVO
Filling
MVC
0
0
50
100
Volume (ml)
150
200
Preload and Afterload
Pressure (kPa)
20
16
control
12
 preload
8
 afterload
4
0
0
50
100
Volume (ml)
150
200
Time-Varying Elastance
P(t) = E(t){V(t) - V0}
E(200)
= Emax
E(160 msec)
Pressure (kPa)
20
E(120 msec)
16
12
E(80 msec)
8
4
0
0
50
100
150
LV Volume (ml)
200
Starling’s Law of the Heart
(The Frank-Starling Mechanism)
Stroke work
increased contractility (e.g.
adrenergic agonist)
decreased contractility
(e.g. heart failure)
“Preload” (EDV or EDP)
Contractility (Inotropic State)
increased contractility (e.g.
adrenergic agonist)
Pressure (kPa)
20
decreased contractility
(e.g. heart failure)
16
12
8
4
0
0
50
100
Volume (ml)
150
200
Physiological Basis of Starling’s
Law
20
Pressure (kPa)
16
12
8
4
0
0
50
100
Volume (ml)
150
200
Overview
• Anatomy and physiology of the heart
– System and organ level
– Resting cardiac tissue
– Active force generation: the sarcomere
• Multi-scale modeling
– Models of cardiac mechanics: from cell to system
– Models of cardiac electromechanics
Fiber and Sheet Architecture
Epicardium
x510
Endocardium
Minimizing Stress Gradients
• Residual Stress
• Fiber Angles
Sarcomere length (µm)
• Torsion
1.94
1.90
1.86
1.82
1.78
1.74
unloaded
Stress-free
Epi
Transmural position
Endo
Resting Tissue Properties
•
Nonlinearity
•
Hysteresis
•
Creep
•
Relaxation
•
Preconditioning Behavior
•
Strain Softening
•
Anisotropy
Passive Biaxial Properties
10
Fiber stress
Stress (kPa)
8
6
4
Cross-fiber stress
2
0
0.00
0.05
0.10
0.15
0.20
Equibiaxial Strain
0.25
Measurement of Myocardial
strain
• Radiopaque beads and biplane x-ray
• video imaging of markers
• ultrasound
• MRI tagging
Myocyte Connections
• Myocytes connect
to an average of 11
other cells (half
end-to-end and half
side-to-side)
• Myocytes branch
(about 12-15º)
• Intercalated disks
– gap junctions
Overview
• Anatomy and physiology of the heart
– System and organ level
– Resting cardiac tissue
– Active force generation: the sarcomere
• Multi-scale modeling
– Models of cardiac mechanics: from cell to system
– Models of cardiac electromechanics
Cardiac Myocytes
•
•
•
•
Rod-shaped
Striated
80-100 m long
15-25 m diameter
Striated Muscle Ultrastructure
Electron micrograph of longitudinal section of freeze-substituted, relaxed
rabbit psoas muscle. Sarcomere shows A band, I band, H band, M line,
and Z line. Scale bar, 100 nm. From Millman BM, Physiol. Rev. 78: 359391, 1998
The Sarcomere
The Sarcomere
Crossbridge Cycle
ExcitationContraction
Coupling
• Calcium-induced
calcium release
• Calcium current
• Na+/Ca2+ exchange
• Sarcolemmal Ca2+
pump
• SR Ca2+ ATPdependent pump
http://www.meddean.luc.edu/lumen/DeptWebs/physio/bers.html
Isometric Tension in Skeletal Muscle:
Sliding Filament Theory
(a) Tension-length curves for
frog sartorius muscle at 0ºC
(b) Developed tension versus
length for a single fiber of
frog semitendinosus muscle
Isometric Testing
Sarcomere
length, m
2.1
Sarcomere isometric
2.0
1.9
Muscle isometric
Tension,
mN
2.0
1.0
time, msec
100 200 300 400
500 600
700
Length-Dependent Activation
2.2 micrometer
1.6 micrometer
Isometric peak twitch tension in cardiac muscle continues to
rise at sarcomere lengths >2 m due to sarcomere-length
dependent increase in myofilament calcium sensitivity
Isotonic Testing
Isovelocity release
experiment conducting
during a twitch
Cardiac muscle forcevelocity relation corrected
for viscous forces of
passive cardiac muscle
which reduce shortening
velocity
Ventricular Mechanics:
Summary of Key Points
• Ventricular geometry is 3-D and complex
• Fiber angles vary smoothly across the wall
• Systole consists of isovolumic contraction and ejection;
diastole consists of isovolumic relaxation and filling
• Area of the pressure-volume loop is ventricular stroke
work which increases with filling (Starling’s Law)
• Ventricles behave like time-varying elastances
• The slope of the end-systolic pressure volume relation is
a load-independent measure of contractility or inotropic
state.
Ventricular Mechanics:
Summary of Key Points (cont’d)
• Collagen contributes to anisotropic resting properties
• Myocardial strain can be measured invasively and noninvasively
• Torsion and residual stress tend to compensate for these
gradients in the ventricles to maintain uniform fiber strain
Overview
• Anatomy and physiology of the heart
– System and organ level
– Resting cardiac tissue
– Active force generation: the sarcomere
• Multi-scale modeling
– Models of cardiac mechanics: from cell to system
– Models of cardiac electromechanics
Integrative In-Silico Biology
Functional Integration, Structural Integration
• Functional integration
– of interacting physiological processes
• Structural integration
– across scales of biological organization
(c) 2004 Andrew McCulloch,
UCSD
Why modeling
• hypothesis generation
• clinical applications
– diagnosis
– training platforms for surgeons
– predict outcomes of surgical interventions
– predict outcomes of therapies
Why multiscale modeling
• Cardiac structure and function are heterogeneous:
most pathologies are regional and nonhomogeneous
• Ca2+ important ion in electrophysiology and
responsible for cardiac force generation
• Many interacting subsystems in basic processes:
e.g.
– ventricular stress  coronary flow
– electrical activation  mechanical activation (ECC and
MEF)
– feedback of baroreceptors on cardiac contractility and
frequency
Overview
• Anatomy and physiology of the heart
– System and organ level
– Resting cardiac tissue
– Active force generation: the sarcomere
• Multi-scale modeling
– Models of cardiac mechanics: from cell to system
– Models of cardiac electromechanics
Models of cardiac mechanics
cellular
• Development of models of cellular cardiac mechanics
have lagged behind models of cellular cardiac
electrophysiology, due to
– lack of available solving algorithms (and computer power)
– controversies about basic mechanisms of force generation in
myofilaments
• 4 categories:
–
–
–
–
phenomenological time-varying elastance models (algebraic)
phenomenological Hill-models (ODE)
A.F.Huxley type models of crossbridge formation (PDE)
Landesberg type myofilament activation model (ODEs)
Modeling Myofilament Force
Production
• Ca2+ binding to TnC
causes tropomyosin to
change to a permissive
state
• Force development
occurs as actin-myosin
crossbridges form
• Crossbridges can ‘hold’
tropomyosin in the
permissive state even
after Ca2+ has
dissociated
Myofilament Activation/Crossbridge
Cycling Kinetics
kn
*
Roff
k*off
kb
Ca2+
R0off
Non-permissive Tropomyosin
Ca2+
kon
koff
Ca2+
0
Ron
*
Ron
Permissive Tropomyosin
Ca2+
g
f
g
f
Ca2+
A10
A1*
Ca2+
Ca2+
bound
to TnC
Ca2+
not
bound
to TnC
Permissive Tropomyosin, 1-3 crossbridges
attached (force generating states)
This scheme is used to find A(t), the timecourse of attached crossbridges for a given
input of [Ca2](t)
Myofilament Model Equations
• Total force is the product of the total number of attached
crossbridges, average crossbridge distortion, and crossbridge
stiffness:
F  At xt 
Myofilament model: results
Noff
0
N*on
R*on
Non
0
+
Ron
0
A*1
A1
A*2
A2
0
0
A3
µtitin
0.5
0
0
0.3
[Ca]
Active Force
SL
ηcell
Simultaneous measurement of
intracellular Ca2+ and shortening in
single myocytes
0.6
Time (s)
µgel
Model validation experiments
•
1
Passive mechanics of
single myocyte
Factive
A*3
[Ca]
Active Force
SL
Myofilament model of
active force generation
Relative Units
Roff
Relative Units
0
R*off
Relative [Ca], Active Force, and SL
0
N*off
1
0.5
0
0
0.3
Time (s)
0.6
Nonlinear Elasticity of Soft
Tissues
•
•
•
•
•
•
Soft tissues are not elastic — stress depends on
strain and the history of strain
However, the hysteresis loop is only weakly
dependent on strain rate
It may be reasonable to assume that tissues in vivo
are preconditioned
Fung: elasticity may be suitable for soft tissues, if we
use a different stress-strain relation for loading and
unloading – the pseudoelasticity concept
a rationale for applying elasticity theory to soft tissues
Unlike in bone, linear elasticity is inappropriate for soft
tissues; we need nonlinear finite elasticity
Transversely Isotropic Laws for
Exponential: Myocardium


W  C e Q  1 Exponential
2
Q  2a1E11  E 22  E33   a2 E11
 

2
2
2
2
2
2
2
2
 a3 E 22
 E33
 E 23
 E32
 a4 E12
 E 21
 E13
 E31
transversely isotropic, x1= "fiber" axis
C=0.6kPa, a1=2.5, a2=15, a3=0, a 4=10
Isotropic + Anisotropic terms:
W  W I1, I 2   W 2 k1 
k1 is the fiber extension ratio, e.g. (Humphrey & Yin, 1987)
 c 4 k112 
c 2I13 
W  c1e
 c 3 e



Polynomial:
W I1,f   C f 1 2  C 2 f  13
 C 3 I1  3   C 4 I1  3 f  1
f  fiber stretch 
 C5 I1  3 2
C1  20, C 2  50, C 3  1.5, C 4  20, C 5  25

Models of cardiac mechanics
tissue
• Passive
– strain energy functions
– orthotropic (fiber – crossfiber – sheet)
– heterogeneous
• Active
– orthotropic (fiber – crossfiber – sheet?)
– heterogeneous(!) (Cordeiro et al, AJP 286, H1471-H1479, 2004)
Models of cardiac mechanics
organ
• Solve tissue models on anatomy with e.g.
finite element or finite difference method
• Compute part of cardiac cycle, e.g.
produce Frank-Starling curves, or
• Compute full cardiac cycle with coupling to
circulatory model
Ventricular Geometry
Truncated ellipses of
revolution
Prolate Spheroidal
Coordinates
x1
x1 = d cosh cos
x2 = d sinh sin cosq
q
x2
b
d
a


x3 = d sinh sin sinq
Anatomic models
Vetter FJ et al (1998) PBMB
Stevens C et al (2003) PBMB
Nielsen PMF et al (1991) AJP Grieshaber J et al (2002)
LeGrice IJ et al (1997) AJP
Rao J et al
Smith NP et al (2000) ABME
Kerckhoffs et al (2003)
Models of cardiac mechanics
organ: application
• Diagnostic measures
• Herz et al used finite
element model of
cardiac ischemia to
generate new
measures for
dyskinetic cardiac
tissue
Herz et al. , 2005. Annals Biomed Eng 33: 912-919
Models of cardiac mechanics
organ: application
• Surgical training platforms
• Predict outcome of surgical interventions
Myosplint reduced
fiber stress, but did
not affect stroke
volume
Guccione et al. 2003. Annals of Thoracic Surgery 76,1171-1180.
Circulatory models
system level
• 2-, 3-, 4-element windkessel
Circulatory models
system level
 windkessel = air chamber
 used in plunger pumps to
ensure a steady flow
Circulatory models
system level
Circulatory models
system level
Pressure serves as hemodynamic boundary condition
Cavity
pressure
Cavity
pressure
Flow Q
 Qdt
FE Cavity volume
(c) 2004 Andrew McCulloch,
UCSD
Cavity volume from
circulatory model
Coupling
  PL 
P 
 PR 
Estimate LV & RV cavity pressure
Circulatory model
FE model
 FE VLFE 
V   FE 
VR 
Circ Cavity volumes
FE Cavity volumes
  FE  circ
R  V V
Calculate difference R
R < criterion?
no
yes
Do not update
Jacobian
 new
P
Update
Jacobian
next timestep


 old R
 P  
 P
 circ VLcirc 
V   circ 
VR 
1
 
 R
p old 

System compliance matrix

 FE
R
V
  
P Pi
P
Pi
 circ
V
 
P
Pi
 VL FE

PL


FE
 VR
 P
 L
FE
circ
VL   VL
 
PR   P L
FE 
circ
VR   VR
PR   P L
circ
VL 

P R 
circ
VR 
P R 
FE compliance matrix Circ compliance matrix
Update 1: Estimate pressure from history
Update 2: Perturb LV pressure
Update 3: Perturb RV pressure
Updates >3: Update pressures
Circulatory models
system level
Overview
• Anatomy and physiology of the heart
– System and organ level
– Resting cardiac tissue
– Active force generation: the sarcomere
• Multi-scale modeling
– Models of cardiac mechanics: from cell to system
– Models of cardiac electromechanics
Important components of
models of cardiac electromechanics
Anatomic model
Hemodynamic
model
Electrophysiology
model
Mechanics
model
time
Models of cardiac electrophysiology
tissue
• Couple ionic models or FitzHugh-Nagumo with
– Monodomain
– Bidomain
Vm(x,t)
Vextr(x,t) and Vintr(x,t)
• Or derive wavefront from bidomain model:
– Eikonal-diffusion tdep(x)
Models of cardiac electromechanics
cellular
• Calcium
• Mechano-electric feedback (MEF)
– deformation
– sarcomere length dependent myofilament calcium sensitivity
– stretch-activated ion channels
• Clinic:
– Resynchronization
– asymmetric hypertrophy by chronic pacing
• Couple existing models of electrophysiology to models of
cardiac mechanics
Models of cardiac electromechanics
tissue
• Electromechanical Coupling:
– Tight (Continuous interplay: EC and MEF)
– Loose (EC only)
• Due to computational demand of tightly
coupled electromechanics, solve models
on 2D domains
Models of cardiac electromechanics
tissue
• Spiral waves with FitzHugh-Nagumomonodomain model and active contraction
• Mechano-electric feedback through
deforming tissue
noncontracting
contracting
Nash and Panfilov. Progr Biophys Mol Biol 85, 501-522, 2004.
Models of cardiac electromechanics
organ
• Solve tissue models on anatomy with e.g.
finite element or finite difference method
• Due to computational demand, either:
– Phenomenological or simple ionic models
– Loosely coupled (ECC only)
• Tightly coupled: parallellization
Models of cardiac electromechanics
organ
• Loosely coupled model
In a computational model study1:
• A physiological sequence of depolarization results in an
unphysiological non-uniform distribution of shortening
• An unphysiological synchronous depolarization results in a
physiological homogeneous distribution of shortening
End-systolic
myofiber strain
0
Experiment (Mazhari et al, Circ. 104, 2001)
Simulation with synchronous activation
-0.1
Simulation with normal activation
-0.2
0
Epi
0.25
0.5
0.75
1
Endo
1Kerckhoffs
et al, ABME 31, p536-547, 2003
Models of cardiac electromechanics
organ
• Loosely coupled model
normal dep wave
synchronous
myofiber
strain
1Kerckhoffs
et al, ABME 31, p536-547, 2003
Models of cardiac electromechanics
organ
From an experimental study1:
The latency to onset of contraction in endocardial cells is
~20 ms longer than that of epicardial cells, which allows
the impulse to traverse the LV wall and effect a
coordinated contraction of the ventricular myocardium.
1Cordeiro
et al, AJP 286, H1471-H1479, 2004
Models ofElectrical
cardiac activation
electromechanics
organ:
pacing
in theventricular
paced heart
Pacing at the right ventricle
activation completed in:
92 ms
Pacing at the left ventricle
129 ms
Kerckhoffs et al. J Eng Math. 2003;47:201-216.
Models
of
cardiac
electromechanics
Contraction in the paced heart
organ: ventricular pacing
Pacing at the right ventricle
Pacing at the left ventricle
strain
Maximum LV
pressure increase:
137 kPa/s
240 kPa/s
Kerckhoffs et al. J Eng Math. 2003;47:201-216.
Models of cardiac electromechanics
FE heart coupled to circulation
Models of cardiac electromechanics
organ
• Tightly coupled model:
– Simple 3-current ionic model
– Model of active contraction
Nickerson et al. Europace 7, 5118-5127, 2005.
Models of cardiac electromechanics
organ: application
• Tightly coupled model
• Computation time: 3 weeks on 8 parallel processors
Nickerson et al. Europace 7, 5118-5127, 2005.
Computational demand
• With advances in biology, simulations remain
large and time-consuming
• Algorithm improvement
• Parallel computing
Computation speed at CMRG